a special case of partial integration
In determining the antiderivative of a transcendental (http://planetmath.org/AlgebraicFunction) function U whose derivative Uβ² is algebraic (http://planetmath.org/AlgebraicFunction), the result can be obtained when choosing in the formula
β«UVβ²πx=UV-β«VUβ²πx |
of integration by parts βVβ²β‘1;β then one has
β«Uπx=β«Uβ 1πx=Uβ x-β«xβ Uβ²πx. |
The functions U in question are mainly the logarithm (http://planetmath.org/NaturalLogarithm2), the cyclometric functions and the area functions.
Examples.
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1.
β«lnxdx=xlnx-β«xβ 1xπx=xlnx-x+C
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2.
β«arcsinxdx=xarcsinx-β«xβ 1β1-x2πx=xarcsinx+12β«-2xβ1-x2πx=xarcsinx+β1-x2+C
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3.
β«arctanxdx=xarctanx-β«xβ 11+x2πx=xarctanx-12β«2x1+x2πx=xarctanx-12ln(1+x2)+C=xarctanx-lnβ1+x2+C
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4.
β«arcoshxdx=xarcoshx-β«xβ 1βx2-1πx=xarcoshx-βx2-1+C
The choiceβ Vβ²β‘1β works as well in such cases asβ β«(lnx)2πxβ andβ β«ln(lnx)πx,β giving respectivelyβ x((lnx)2-2lnx+2)+Cβ andβ xln(lnx)-Lix+C (see logarithmic integral). Alsoβ
β«(arcsinx)2πxβ , requiring two integrations by parts, and giving the resultβ x(arcsinx)2+2β1-x2arcsinx-2x+C.
Title | a special case of partial integration |
---|---|
Canonical name | ASpecialCaseOfPartialIntegration |
Date of creation | 2013-03-22 17:38:35 |
Last modified on | 2013-03-22 17:38:35 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Feature |
Classification | msc 26A36 |
Related topic | IntegralTables |